Equivalence Relations

Relation is a branch of mathematics which plays a very important role. Let P and Q are two Sets then a relation R defined from Set P to set Q, is a subset of P X Q. Relation can be denoted in the list form and in tabular form as well, for example a relation ‘R’ on set P = 1, 2, 3, 4, 5 defined by R = (P, Q): Q = P + 2 can also be expressed as:-
P R Q if and only if Q = P + 2,
Let ‘U’ and ‘V’ the two Sets. Then a relation ‘R’ from set ‘U’ to set ‘V’ is a subset of U x V. Thus ‘R’ is a relation from U to V ↔ R is subset U x V. If ‘R’ is a relation from a non void set ‘U’ to a non void set ‘V’ and if (u, v) ε R, then we write u R v, which is read as 'u’ related to ‘v’ by the relation ‘R'. If (u, v) ε R, then we write ‘u ! R v’ and we can say that ‘u’ is not related to ‘v’ by the relation ‘R’.
Let’s see the types of relation which are shown below:
-Void, Universal, And Identity Relation.
-Reflexive Relation.
-Symmetric Relation.
-Anti Symmetric Relation.
-Transitive Relation.
A relation R on a set P is said to be an equivalence Relations on set P if and only if the given relation should be reflexive, symmetric and transitive. If the relation follows all these three condition then the relation is equivalence relations.
1.     Reflexive relation i.e. (P, P) ∊R got all P ∊R.
2.     Symmetric relation i.e. (P, Q) ∊then (Q ∊R) for all a, b ∊R.
3.     Transitive relation i.e. (P, Q) ∊R and (Q, R) ∊R then (P, R) ∊R for all P, Q, R ∊R.
This is all about equivalence relations.

Partial Orders

Partial orders can be defined as orders which show a relation 'R', over a Set 'S', partial order relation is represented by ≤.  Any relation 'R', is a partial ordering only if it will satisfy following three conditions.
1.     Reflexivity: x ≤ y.
2.     Antisymmetry: x ≤ y and y ≤ x which implies x = y.
3.     Transitivity: x ≤ y and y ≤ z implies x≤ z.